Group invariant Colombeau generalized functions

arXiv:math/0512219v2 [math.FA] 16 Nov 2006

Group invariant Colombeau generalized functions Hans Vernaeve∗ Institut f¨ ur Grundlagen der Bauingenieurwissenschaften Technikerstraße 13 A-6020 Innsbruck

Abstract Colombeau generalized functions invariant under smooth (additive) one-parameter groups are characterized. This characterization is applied to generalized functions invariant under orthogonal groups of arbitrary signature, such as groups of rotations or the Lorentz group. Further, a one-dimensional Colombeau generalized function with two (real) periods is shown to be a generalized constant, when the ratio of the periods is an algebraic nonrational number. Finally, a nonstandard Colombeau generalized function invariant under standard translations is shown to be constant.

Key words: Colombeau generalized functions, translation invariance, rotational invariance, Lorentz invariance, generalized one-parameter groups. 2000 Mathematics subject classification: 46F30, 35D05.

1

Introduction

This paper is related to a series of papers on group invariant generalized functions that appeared during the last years ([7], [8], [9], [10], [13]). In particular, this paper focuses on a type of questions that remained an open problem for several years: if a generalized function is invariant under all ∗

Supported by research grant M949 of the Austrian Science Foundation (FWF)

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non-generalized transformations of a generalized transformation group, is it then invariant under the whole (generalized) group? Only recently, the key case of a translation group was solved in the affirmative in [14]. In [7], it is already shown that this result can be applied to solve the question for the group of rotations. In this paper, we show that it can be applied to more general groups as well. We prove a general result on invariance under smooth one-parameter groups (section 3) and indicate how it can be applied to invariance under various matrix groups (sections 4, 5). In the case of rotations, the same characterization as in [7] is obtained. This development mirrors results on group invariant Schwartz distributions, following the work by Schwartz [15] (see [2], [6], [11], [16], [17] and references therein). In section 6, we revisit the case of the translation groups. We give two more proofs of this theorem. Doing so, the following new results are obtained: a one-dimensional Colombeau generalized function with two (real) periods is a generalized constant, when the ratio of the periods is an algebraic nonrational number; a nonstandard Colombeau generalized function invariant under standard translations is constant.

2

Preliminaries

We work in the (so called special) Colombeau algebra G(Rd ) of generalized functions on Rd (d ∈ N), defined as follows [4, 5]. Denote by E(Rd ) the algebra of all nets (uε )ε∈(0,1) of C ∞ -functions Rd → C. Then G(Rd ) = EM (Rd )/N (Rd ), where  EM (Rd ) = (uε )ε ∈ E(Rd ) : (∀K ⊂⊂ Rd )(∀α ∈ Nd )(∃b ∈ R)
sup |∂ α uε (x)| = O(εb), as ε → 0 x∈K  d N (R ) = (uε )ε ∈ E(Rd ) : (∀K ⊂⊂ Rd )(∀α ∈ Nd )(∀b ∈ R)
sup |∂ α uε (x)| = O(εb), as ε → 0 . x∈K

We recall a result about the composition of generalized functions. Definition 1. An element (fε )ε = (f1,ε , . . . , fd,ε ) ∈ EM (Rd )d is called cbounded if (∀K ⊂⊂ Rd )(∃K ′ ⊂⊂ Rd )(∃ε0 > 0)(∀ε < ε0 )(fε (K) ⊆ K ′ ). 2

An element of G(Rd )d is called c-bounded if it possesses a c-bounded representative. Lemma 1. 1. Let f ∈ G(Rd )d be c-bounded and g ∈ G(Rd ). Then the composition g ◦ f defined on representatives by means of (g ◦ f )ε = gε ◦ fε is a well-defined generalized function in G(Rd ). 2. Let f , g ∈ G(Rd )d be c-bounded. Then g ◦ f (similarly defined on representatives) is a well-defined c-bounded generalized function in G(Rd )d . Proof. 1. See [5, section 1.2.1]. 2. The well-definedness follows directly from the first part; the c-boundedness of g ◦ f follows directly from (g ◦ f )ε (K) ⊆ gε (fε (K)) and the c-boundedness of f and g. In the last section, we will also work in the algebra of nonstandard Colombeau generalized functions ρ E(Rd), defined as follows [12, 18]. Let ρ ∈ ∗R be a fixed positive infinitesimal. For x, y ∈ ∗Cd , we write x ≅ y iff |x − y| ≤ ρn , ∀n ∈ N (x − y is then called a iota, or negligible). For x ∈ ∗Cd , we write x ∈ ∗CM iff |x| ≤ 1/ρn for some n ∈ N (x is then called moderate). We denote by ns(∗Rd ) the set of near-standard (=finite) elements of ∗Rd . Then ρ E(Rd ) = EM (Rd )/N (Rd), where EM (Rd ) = {u ∈ ∗C ∞ (Rd ) : (∀x ∈ ns(∗Rd ))(∀α ∈ Nn )(∂ α u(x) ∈ ∗CM )} N (Rd ) = {u ∈ ∗C ∞ (Rd ) : (∀x ∈ ns(∗Rd ))(∀α ∈ Nn )(∂ α u(x) ≅ 0)}.

3

Invariance under one-parameter groups

Theorem 2. Let gθ be a one-parameter (additive) group action on Rd , i.e. 1. gθ is a bijection, ∀θ ∈ R 2. gθ1 +θ2 = gθ1 ◦ gθ2 , ∀θ1 , θ2 ∈ R 3. g0 is the identity mapping on Rd 3

4. g−θ = (gθ )−1 , ∀θ ∈ R. Suppose further that the map (θ, x) 7→ gθ (x) is C ∞ . e c , f ◦ gθ is well-defined. Let f ∈ G(Rd ). Then for each θ ∈ R e c , where R e c is the ring of If f ◦ gθ = f , ∀θ ∈ R, then f ◦ gθ = f , ∀θ ∈ R Colombeau generalized numbers with bounded representative. Proof. First, notice that the C ∞ -character of (θ, x) 7→ gθ implies that k   ∂ α d (∀k ∈ N)(∀α ∈ N )(∀R ∈ R) sup sup k ∂ gθ (x) < +∞ |θ|≤R |x|≤R ∂θ

(1)

e c and so in particular, gθ is a c-bounded generalized function, for each θ ∈ R the composition is well-defined. e d . Define h ∈ G(R) by means of h(θ) = f (gθ (a)), i.e., by a Fix a = (aε )ε ∈ R c definition on representatives, hε (θ) = fε (gθ (aε )). By the c-boundedness of gθ , one sees that this definition is independent of the representative of f . Further, by equation (1), one also sees that (hε )ε ∈ EM (R). Now let Tt be the translation-operator x 7→ x + t on R. Then (h ◦ Tt )(θ) = f (gθ+t (a)) = (f ◦ gt )(gθ (a)), so by the hypothesis on f (h ◦ Tt )(θ) = f (gθ (a)) = h(θ). So h is a generalized constant (see section 6). I.e., for each C ∈ R+ ,   + p (∀p ∈ N)(∃ε0 ∈ R )(∀ε < ε0 ) sup |hε (θ) − hε (0)| ≤ ε . θ∈R |θ| εN n.

x∈K

Now let fn := uεn − uεn (0). So there exists a sequence (an )n∈N , an ∈ K such that |fn (an )| > εN n, for each n. We have kan k < C, for some C ∈ R+ . Now let n  o \ εN n d An := x ∈ R : |fn (x)| < , Bn := Am . 3 m≥n

As fn (0) = 0, ∀n, 0 ∈ B0 . Let x ∈ Rd . Because of the translation-invariance, also x ∈ Bn , for some n. Similarly, we define n  \ εN o , Dn := Cm . Cn := x ∈ Rd : |fn (x + an ) − fn (an )| < n 3 m≥n Let x ∈ Rd . Again, x ∈ Dn , for some n. Both (BnS), (Dn ) are increasing sequences of measurable subsets of Rd and S Bn = Dn = Rd . We denote the Lebesgue measure by µ and the open ball with center x ∈ Rd and radius r ∈ R+ by B(x, r). Now µ(B(0, C) \ Bn ) → 0, µ(B(0, 2C) \ Dn ) → 0 as n → ∞. Since Bn ⊆ An , Dn ⊆ Cn , also µ(B(0, C) \ An ) → 0, µ(B(0, 2C) \ Cn ) → 0 as n → ∞. Finally, let n  εN o En := x ∈ Rd : |fn (x) − fn (an )| < n = Cn + an . 3 Then µ(B(0, C) \ En ) ≤ µ(B(an , 2C) \ En ) = µ(B(0, 2C) \ Cn ) → 0, since kan k < C. So µ(B(0, C) \ (An ∩ En )) → 0. In particular, An and En have a non-empty intersection as soon as n is large enough. Clearly, this is impossible. 9

The second proof is for the one-dimensional case (the more-dimensional case can then be obtained e.g. in a similar way as we obtained theorem 4 by applying theorem 2, or as in the corollary to theorem 13) and uses weaker hypotheses on the generalized function. Moreover, both the previous proof and the proof given in [14] cannot (at least not a priori) be generalized to the nonstandard case (the ultrafilter destroys the argument). Although the proof of the nonstandard theorem is a little more conceptual, the proof of the standard version doesn’t make use of nonstandard analysis. We recall two number theoretic theorems. Theorem 8 (Dirichlet’s approximation theorem). Let α ∈ R+ \ Q. Then   1 (∀N ∈ N \ {0})(∃k, l ∈ N) 0 < l ≤ N & |k − lα| ≤ . N Proof. See [1]. Theorem 9 (Liouville’s approximation theorem). Let α ∈ R+ \ Q be an algebraic number n. Then there exists c ∈ R+ such that for each k, of kdegree l ∈ N (l 6= 0), α − l ≥ lcn . Proof. See [1].

For our application, we will use the following corollary: Corollary. Let α ∈ R+ \Q be an algebraic number. Then there exists M ∈ N such that   2 1 . ≤ |k − lα| ≤ (∀R ∈ (2, +∞))(∃k, l ∈ N) l ≤ R & RM R Proof. Let R ∈ R, R ≥ 2. Let N ∈ N such that R − 1 ≤ N ≤ R. Then by Dirichlet’s approximation theorem, there exist k, l ∈ N such that 0 < l ≤ R 1 and |k − lα| ≤ R−1 ≤ R2 . By Liouville’s approximation theorem, there exist c ∈ R+ and n ∈ N (n ≥ 2, both depending on α only) such that |k − lα| ≥ c c ≥ Rn−1 ≥ R1M , for a good choice of M (depending on c and n, hence on ln−1 α only). We still need the following elementary lemma.

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Lemma 10. Let f : R → C be almost-periodic on an interval [a, b] with periods h1 ,h2 ∈ R+ and tolerance ε ∈ R+ , i.e., for i = 1, 2 (∀x ∈ [a, b])(x + hi ∈ [a, b] =⇒ |f (x + hi ) − f (x)| ≤ ε). Then (∀k, l ∈ N)(∀x ∈ [a + h1 + h2 , b − h1 − h2 ]) (x + kh1 − lh2 ∈ [a, b] =⇒ |f (x + kh1 − lh2 ) − f (x)| ≤ (k + l)ε). Proof. Let x ∈ [a + h1 + h2 , b − h1 − h2 ]. Then the value of f differs at most by ε every time we move with a step ±h1 or ±h2 , as long as we ensure that all points lie in [a, b]. So it is sufficient to ensure that we reach x + kh1 − lh2 in at most k + l such steps. We take the following steps: x + h1 , x + 2h1 , . . . , x + k ′ h1 , where k ′ is the largest number such that x + k ′ h1 ≤ b (by the hypotheses, at least one step is taken). Then we move to x + k ′ h1 − h2 , x + k ′ h1 − 2h2 , . . . , x + k ′ h1 − l′ h2 , where l′ is the largest number such that x + k ′ h1 − l′ h2 ≥ a (again by the hypotheses, at least one step is taken). Repeating this procedure, the coefficients of h1 and h2 increase until either the coefficient of h1 equals k or the coefficient of h2 equals l. In the first case, we have x + kh1 − l′ h2 ∈ [a, b] with 0 ≤ l′ ≤ l. By hypothesis, also x + kh1 − lh2 ∈ [a, b], so all the remaining steps x + kh1 − (l′ + 1)h2 , . . . , x + kh1 + (l − 1)h2 also lie in [a, b]. The second case is similar. Theorem 11. Let f ∈ G(R). Let α ∈ R+ \ Q be an algebraic number. If f is periodic with periods 1 and α, i.e., if f (x + 1) = f (x) and f (x + α) = f (x) hold as equalities in G(R), then f is a generalized constant. Proof. Let p ∈ √ N and K = [−R, R] ⊂⊂ R. By the corollary to thm. 9, we find ∀ε ∈ (0, 1/ p 2), some kε and lε ∈ N, lε ≤ 1/εp such that εM p ≤ |kε − αlε | ≤ 2εp . In particular, kε ≤ αlε + 2εp ≤ (α + 1)/εp . By the hypothesis, there exists ε0 > 0 such that for each ε ≤ ε0 , fε is almostperiodic on K with periods 1, α and tolerance ε(M +2)p . By lemma 10, we conclude that for x ∈ R with |x| ≤ R − α − 1, |fε (x + kε − αlε ) − fε (x)| ≤ (kε + lε )ε(M +2)p , 11

since |x + kε − αlε | ≤ |x| + 2εp ≤ R. Let hε := |kε − αlε |. Now for each x ∈ R with |x| ≤ R − α − 2, we can find λε ∈ Z with |λε − x/hε | ≤ 1, so also |λε hε − x| ≤ 2εp . Then for each j ∈ Z with |j| ≤ |λε |, |jhε | ≤ R − α − 1, so |fε (λε hε ) − fε (0)| ≤

|λε | X j=1

|fε (σλε jhε ) − fε (σλε (j − 1)hε )|

≤ |λε | (kε + lε )ε

(M +2)p

≤



 |x| + 1 (α + 2)ε(M +1)p ≤ cεp , hε

for some constant c only depending on R (here σλε = ±1 is the sign of λε ). By the moderateness of f ′ , there exists N ∈ N (only depending on R) such that |fε (x) − fε (λε hε )| ≤ |x − λε hε | ε−N ≤ 2εp−N as soon as ε ≤ ε0 (possibly with a smaller ε0 ). So sup |x|≤R−α−2

|fε (x) − fε (0)| ≤ cεp + 2εp−N ,

ε ≤ ε0 .

As R and p are arbitrary, it follows that f is a generalized constant. Corollary. Let f ∈ G(R). If f is periodic with periods h1 and h2 , with h1 /h2 ∈ R+ \ Q algebraic, then f is a generalized constant. Question: suppose f ∈ G(R) is periodic with periods h1 , h2 ∈ R+ and suppose that h1 /h2 ∈ / Q. Is f a generalized constant? A generalization of the corollary to thm. 9 for arbitrary h ∈ R+ \ Q instead of algebraic numbers is sufficient. Notice that an approximation as in Liouville’s approximation theorem, and hence also the corollary, holds for many transcendent numbers as well [3] (the exceptions form a set of Hausdorff dimension 0). We conclude with the nonstandard version. Lemma 12. Let Ω ⊆ Rd , Ω open. For an internal map f : ∗Ω → ∗C, the following are equivalent: 1. (∀x, y ∈ ns(∗Ω))(x ≅ y =⇒ f (x) ≅ f (y)) 2. (∀K ⊂⊂ Ω)(∀x ∈ ∗K)(∀n ∈ N)(∃m ∈ N)(∀y ∈ ∗K)(|x − y| < ρm =⇒ |f (x) − f (y)| < ρn ) In such case, we say that f is ≅-continuous. 12

Proof. ⇒: Let K ⊂⊂ Ω, x ∈ ∗K and n ∈ N. By underspill, the internal set {m ∈ ∗N : (∀y ∈ ∗K)(|x − y| < ρm =⇒ |f (x) − f (y)| < ρn )} contains some m ∈ N. ⇐: Let x, y ∈ ns(∗Ω). Then there exists K ⊂⊂ Ω such that x, y ∈ ∗K. If x ≅ y, we have |x − y| < ρm , ∀m ∈ N, so the hypothesis learns that (∀n ∈ N)(|f (x) − f (y)| < ρn ), i.e., f (x) ≅ f (y). Theorem 13. Let f : ∗R → ∗C be internal and ≅-continuous. Let α ∈ R+ \Q be an algebraic number. Suppose that for h ∈ {1, α}, f is almost-periodic up to iotas with period h, i.e., (∀x ∈ ns(∗R))(f (x + h) ≅ f (x)). Then f is constant up to iotas, i.e., (∀x ∈ ns(∗R))(f (x) ≅ f (0)). Proof. By transfer on the corollary to thm. 9, we find for each n ∈ N some k, l ∈ ∗N, l ≤ 1/ρn , such that (still with the same standard M) ρnM ≤ |k − αl| ≤ 2ρn . In particular, k, l are moderate. Let R ∈ R+ . By overspill, the almost-periodicity up to iotas implies that there exists some ω ∈ ∗N \ N such that for h ∈ {1, α} (∀x ∈ ∗R)(|x| ≤ R =⇒ |f (x + h) − f (x)| ≤ ρω ). By transfer on lemma 10, (∀k, l ∈ ∗N)(∀x ∈ ∗[ − R + α + 1, R − α − 1]) (|x + k − lα| ≤ R =⇒ |f (x + k − lα) − f (x)| ≤ (k + l)ρω ). If k, l are moderate and k − lα ≈ 0, we have in particular (as R can be taken arbitrarily large) that k − lα is an almost-period up to iotas for f on the whole of ns(∗R). So for each n ∈ N, f has almost-periods up to iotas hn with ρnM < hn < 2ρn . 13

Since hn 6≅ 0, we find for each x ∈ ns(∗R) a moderate λ ∈ ∗Z such that |λhn − x| < 2ρn . Then |f (λhn ) − f (0)| ≤

|λ| X j=1

|f (σλ jhn ) − f (σλ (j − 1)hn )| ,

so f (λhn ) ≅ f (0) (here σλ = ±1 is the sign of λ). Since f is internal and ≅-continuous, by the second characterization in lemma 12, this implies that f (x) ≅ f (0). Corollary. Let f ∈ ρ E(R). If f is periodic with periods h1 and h2 , with h1 /h2 ∈ R+ \ Q algebraic, then f is constant. Corollary. Let f ∈ ρ E(Rd). If f is invariant under standard translations, i.e., f (x + h) = f (x) holds in ρ E(Rd ), ∀h ∈ Rd , then f is constant. Proof. Let f˜ ∈ ∗C ∞ (Rd ) be a representative of f and let a = (a1 , . . . , ad ) ∈ ns(∗Rd ). Let g˜(t) = f˜(a1 , . . . , ad−1 , t), t ∈ ∗R. Then g˜ ∈ ∗C ∞ (R) is a representative of an element g ∈ ρ E(R). For standard h ∈ R and t ∈ ns(∗R), g˜(t+h) = f˜(a1 , . . . , ad−1 , t + h) ≅ f˜(a1 , . . . , ad−1 , t) = g˜(t), as f is invariant under standard translations. This means that g is invariant under standard translations, and is constant. In particular, f˜(a) = g˜(ad ) ≅ g˜(0) = f˜(a1 , . . . , ad−1 , 0). Similarly, we obtain f˜(a) ≅ f˜(a1 , . . . , ad−1 , 0) ≅ f˜(a1 , . . . , ad−2 , 0, 0) ≅ · · · ≅ f˜(0). As a ∈ ns(∗Rd ) arbitrary, f is constant.

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